On the best polygonal approximation of fuzzy numbers by using a family of metrics based on mid-spread formulation

Polygonal fuzzy numbers are an important family of fuzzy numbers with a wide range of applications in several fields in sciences and engineering. The simplicity of polygonal fuzzy numbers provides very efficient ways of computations and allows us to have a simple interpretation of the phenomena described by the fuzzy number. In this paper, we address the problem of finding the best polygonal fuzzy number for a given fuzzy number by using a general family of metrics formulated through the mid-spread representation of fuzzy numbers. These metrics are an extension of the previous ones as they take into account not only the distance between the extreme values of the fuzzy numbers but also the distance between other representative elements of the fuzzy numbers related to the corresponding α-cuts through an adequate weight mapping. By using a novel suitable matrix representation of polygonal fuzzy numbers, we show that the best polygonal approximation of a fuzzy number is equivalent to a strict convex quadratic optimization problem with linear inequality constraint over a finite-dimensional space. Furthermore, we analyze the stability of the best polygonal approximation, and we give computable sufficient conditions to ensure the uniqueness of the solution for two different weight mappings. We apply our results to the best trapezoidal approximation problem.